Below is a working example of how to achieve this.
You need to reference Microsoft.TeamFoundation.Work.WebApi
.
public async Task<List<TeamSettingsIteration>> GetProjectIterations(string serverUrl, string projectName)
{
var uri = new Uri(serverUrl);
var creds = new VssClientCredentials(new WindowsCredential(true), new VssFederatedCredential(true), CredentialPromptType.PromptIfNeeded);
var azureDevopsConnection = new VssConnection(uri, creds);
await azureDevopsConnection.ConnectAsync();
WorkHttpClient azureDevOpsWorkHttpClient = azureDevopsConnection.GetClient<WorkHttpClient>();
TeamContext teamContext = new TeamContext(projectName);
List<TeamSettingsIteration> results= await azureDevOpsWorkHttpClient.GetTeamIterationsAsync(teamContext);
return results;
}